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HIGH RESOLUTION SURFACE GEOMETRY AND ALBEDO BY COMBINING LASER ALTIMETRY AND VISIBLE
IMAGES
Robin D. Morris, Udo von Toussaint and Peter C. Cheeseman,
NASA Ames Research Center, MS 269-2, Mo ett Field, CA 94035
[rdm,udt,cheesem]@email.arc.nasa.gov
KEY WORDS: Bayesian inference; surface geometry; albedo; computer vision;
ABSTRACT
The need for accurate geometric and radiometric information over large areas has become increasingly important. Laser altimetry is one of the key technologies for obtaining this geometric information. However, there are important application areas
where the observing platform has its orbit constrained by the other instruments it is carrying, and so the spatial resolution
that can be recorded by the laser altimeter is limited. In this paper we show how information recorded by one of the other
instruments commonly carried, a high-resolution imaging camera, can be combined with the laser altimeter measurements to
give a high resolution estimate both of the surface geometry and its re ectance properties. This estimate has an accuracy
unavailable from other interpolation methods. We present the results from combining synthetic laser altimeter measurements
on a coarse grid with images generated from a surface model to re-create the surface model.
RE SUME
Le besoin d'informations geometriques et radiometriques precises couvrant de grandes etendues devient de plus en plus important. L'altimetrie laser est une des technologies principales pour obtenir ces informations geometriques. Cependant, il
est des domaines d'application importants ou la plateforme d'observation a son orbite contrainte par les autres instruments
qu'elle porte, ce qui limite la resolution spatiale qui peut ^etre enregistree par l'altimetre. Dans cet article nous montrons
comment l'information enregistree par un des autres instruments communement embarques, une camera photographique a
haute resolution, peut ^etre combinee avec les mesures de l'altimetre laser pour donner une estimation haute resolution a la
fois de la geometrie de surface et de ses proprietes de re ectivite. Cette evaluation o re une exactitude inegalee par d'autres
methodes d'interpolation. Nous presentons les resultats obtenus en combinant des mesures synthetiques d'altimetre laser sur
une grille grossiere avec des images produites a partir d'un modele de surface pour recreer le modele de surface.
KURZFASSUNG
Prazise geometrische und radiometrische Informationen uber grosse Areale ist zunehmend von Bedeutung. Die Laser Altimetrie
ist eine der Schlusseltechnologien zur Gewinnung dieser Daten. Allerdings ist in wichtigen Anwendungsfallen die Laser Altimetrie Messung durch weitere Instrumente behindert und daher die raumliche Au osung eingeschrankt. In dieser Vero entlichung
zeigen wir auf wie die von einer hochau osenden Kamera (einer fast immer installierten Diagnostik) gewonnenen Bilder mit
den Daten der Laser Altimetrie kombiniert werden konnen um eine prazise Bestimmung der Ober achenform und ihrer Re ektivitatseigenschaften zu ermoglichen. Diese Art der Ober achenbestimmung erweist sich einer Splineinterpolationen der Laser
Altimetriedaten uberlegen. Wir zeigen die Ergebnisse der Ober achenrekonstruktion aus der Kombination von synthetischen,
niedrig aufgelosten Laser Altimetriedaten und Bildern.
1 INTRODUCTION
tical imager. These images have been previously used to infer a surface reconstruction, solving this inverse problem using
Bayesian probability theory (Smelyanskiy 2000, Morris 2001).
The accuracy of the reconstruction of the 3-dimensional surface depends on the geometric information content of the images and on additional prior knowledge. Often images from
mapping orbits do not contain much geometric information
as the baseline is very small compared to the distance to the
surface.
The need for accurate geometric information for a variety of
problems has grown rapidly in the last decades. These needs
cover a broad eld, from monitoring of environmental changes
such as the deformation rates of glaciers, to the creation of
3-dimensional digital city models, and the determination of
the shapes of asteroids and features on planets. The demands
with respect to the required accuracy are steadily increasing
(Rees 1990).
Laser altimetry systems have been able to respond to these
demands. However, for many applications, only coarse resolution sampling is available. This is especially true for planetary
and small body observations, where the sampling of the surface is constrained by the orbit of the sensor, and this orbit
is often determined by the other instruments carried by the
spacecraft.
These other instruments usually include a high-resolution op-
In this paper we show that a dense surface geometry estimate
can be made by combining the information from a coarse but
highly accurate grid of height eld points from Laser altimetry
measurements and the limited geometrical information from
a set of optical images. The resulting surface estimate (both
geometry and albedo) has a precision unavailable from other
interpolation methods. At the same time far fewer images
are needed for a surface reconstruction than without the data
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106
ing these synthetic observations is straightforward. We also
assume that the error in these measurements are known.
from the laser altimetry measurements.
The calculation is a two step process: Using the images and
a spline interpolation of the laser altimetry data, an approximate albedo eld of the surface is inferred. This albedo eld
and the spline interpolated surface are the starting points for
the Bayesian surface reconstruction. The varying accuracy
of the height eld points is taken into account by assigning
di erent uncertainty values to the individual points. The uncertainty of the laser altimetry measured points is very much
lower compared with the interpolated values. This approach
also o ers an easy way to combine measurements with di erent accuracy. The height eld points between the grid points
of the laser altimetry measurements are updated by the additional geometrical constraints of the optical images. We
present results of the inference of surface models from simulated height eld grids and aerial photographs. The in uence
of di erent number of images and varying grid resolution is
shown.
3 A BAYESIAN FRAMEWORK
In this paper the surface geometry is represented by a triangular mesh and the surface re ectance properties (albedos)
are associated with the vertices of the triangular mesh. We
will consider the case of Lambertian surfaces. We will also
assume that the camera parameters (position and orientation,
and internal calibration) and the parameters of the lighting
are known. It is possible to estimate these parameters in a
similar Bayesian framework, but it is beyond the scope of this
paper (Morris 2001, Smelyanskiy 2001).
Thus we represent the surface model by the pair of vectors
[~z ~ ]. The components of these vectors correspond to the
height and albedo values de ned on a regular grid of points
[~z ~ ] = f(zi ; i ) ; i = ` (q x^ + p y^ )g q; p = 0; 1; : : : (1)
where ` is the elementary grid length, x^ , y^ are an orthonormal
2 THEORY
pair of unit vectors in the (x,y) plane and i indexes the position in the grid. The pair of vectors of heights and albedos
represents a full vector for the surface model
The objective here is to infer a surface model using the
available data, in this case, laser altimeter measurements.
Bayesian inference has, for some time now, been the method
of choice for many inference problems, enabling accurate estimation of parameters of interest from noisy and incomplete
data (Bernardo 1994). It also provides a consistent framework for the incorporation of multiple, distinct, data sets into
the inference process. The general approach is illustrated in
gure 1. The gure shows that synthetic observations of the
model are made using a computer simulation of the observation process, and that these are compared with the actual
observations. The error between the actual and the simulated
observations is used to adjust the parameters of the model, to
minimize the errors. Bayes theorem tells us directly how much
weight to assign to the two sources of errors, those coming
from the image measurements and those coming from the
laser altimeter measurements.
The surface model we use here is a triangulated mesh. At
each vertex of the mesh we store the height and the albedo.
As discussed above, to be able to infer the surface heights
and albedos, we must rst be able to simulate the data that
would be recorded from the surface.
Generating images from the surface model is the area of computer graphics known as rendering (Foley 1990). It is important to note, however, that much recent work in computer
graphics is unsuitable for our purpose, as it works in image
space, where the fundamental unit is the image pixel, and any
given pixel is coloured by light from one and only one surface
element. This results in artefacts due to the relative sizes of
the projections of the surface elements onto the image plane
and their discretization into pixels. These artefacts are particularly noticeable along the edges of the surface elements
(aliasing). For this work we require the renderer to operate
in object space, and below we will brie y describe such a system. We also note that an object-space renderer can also
compute the derivatives of the pixel values with respect to
the surface model parameters. This is crucial in enabling efcient estimation of the surface model parameters, and will
be described in more detail below.
We are also required to produce synthetic laser altimeter measurements. We make the approximation that the laser altimeter makes point measurements of the surface, and so produc-
u = [~z ~ ]:
(2)
To estimate the values of ~z; ~ from the laser altimeter and
image data, we apply Bayes theorem which gives
p(~z; ~ jL; I1 : : : IF ) / p(L; I1 : : : IF j~z; ~ ) p(~z; ~ ); (3)
where L is the laser altimeter data If (f = 1; : : : ; F ) is the
image data. This states that the posterior distribution of the
heights and the albedos is proportional to the likelihood {
the probability of observing the data given the heights and
albedos { multiplied by the prior distribution on the heights
and albedos.
Given the surface description, the images and the laser altimeter measurements are conditionally independent, and equation 3 can be written as
p(~z; ~ jL; I1 : : : IF ) / p(Lj~z; ~ ) p(I1 : : : IF j~z; ~ ) p(~z; ~ );
where we now have two independent likelihood terms, one for
each data stream.
The prior distribution is assumed to be Gaussian
p(~z; ~ ) / exp ; 21 u;1 uT ;
^ h2 0 Q=
;
1
= 0 Q=
^ 2 ;
(4)
where the vector of the surface model parameters u is dened in (2). The inverse covariance matrix is constructed to
enforce a smoothing constraint on local variations of heights
and albedos. We penalize the integral over the surface of the
2
curvature factor c(x; y) = zxx
+ zyy2 + 2zxy2 , and similarly
for albedos. The two hyperparameters h and in equation (4) control the expected values of the surface-averaged
curvatures for heights and albedos.
This prior is placed directly over the height variables, z, but
albedos are only de ned over the range [0 ; 1]. To avoid this,
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MODEL OF
IMAGER
SYNTHETIC
IMAGES
SURFACE
MODEL
107
REAL
IMAGES
BAYES
THEOREM
SYNTHETIC
LASER ALTIMETER
MEASUREMENTS
MODEL OF LASER
ALTIMETER
REAL LASER
ALTIMETER
MEASUREMENTS
Figure 1: Outline of the Bayesian approach to surface reconstruction from images and laser altimeter measurements
we use transformed albedos 0i in the Gaussian (4), where 0i
are de ned by:
where l0 is the vector of actual laser altimeter observations,
and l are the corresponding entries taken from the ~z vector.
The inverse covariance matrix l;1 is a diagonal matrix with
1=l2 on the leading diagonal.
The term for the image measurements is more complex, as
I^(~z; ~) is the rendering process. To make progress with minimizing L(~z; ~) we linearize I^(~z; ~ ) about an initial estimate,
0i = log(i =(1 ; i )); u ! [~z ~0 ]:
(5)
In the vector of model parameters u values of ~ are replaced
by values of ~0 .
For both the likelihoods we make the usual assumption that
the di erences between the observed data and the data synthesized from the model have a zero mean, Gaussian distribution. So for the laser altimeter measurements we have
P
p(Lj~z; ~) / exp ;
l
(Ll ; L^ l (~z; ~ ))
2l2
2
~z0 ; ~0
I^f p ; @ I^f p
I^(~z; ~ ) = I^(~z0 ; ~0 ) + D x; D @@z
@0
i
i
p(I1 : : : IF j~z; ~ ) / exp ;
(I ; I^f p (~z; ~ ))
2e2
2
f;p f p
L0 = 12 x A^ x ; b x; x u ; u0 ;
T
^ ;1
A^ = ;1 + DD
e2 + l ;
^
b = (I ; I (~ze20 ; ~0 )) D
!
L(~z; ~ ) /
(I ; I^f p (~z; ~ ))2
f;p f p
e2
2
^
+ l (Ll ;L2 l (~z; ~ ))
l
+ x ;1 xT ;
P
(7)
where x = u ; u0 is a deviation from a current estimate u0 .
L is a nonlinear function of ~z; ~ and the MAP estimate is
that value of ~z; ~ which minimizes L(~z; ~ ).
The crux of the problem is thus how to minimize L. We apply
a gradient method, using an initialization based on a spline
interpolation of the laser altimeter measurements.
Making the assumption that the laser altimeter makes point
measurements of one of the vertices of the mesh, equation 6
can be written as
;
p(Lj~z; ~) / exp ;1=2(l ; l0 );l 1 (l ; l0 )T
(8)
(9)
where ^ ;l 1 is now a large square matrix (of dimension
length(~z) + length(~)), where the diagonal elements corresponding to the vertices for which there are laser altimeter
measurements take values 1=l2 and all other entries are zero.
The entries of u0 corresponding to the laser altimeter measurements are set to the observed values, and the remaining
height values are initialized using a spline interpolation. This
interpolation and the albedo initialization will be described
below.
In equation 9 A^ is the Hessian matrix of the quadratic form
and vector b is the gradient of the likelihood L computed at
the current estimate. We search for the minimum in x using
a conjugate-gradient method (Press 1992).
Thus the most dicult part of nding the MAP is the requirement to render the image and compute the derivatives
for any values of the surface model parameters. We discuss
this computation in some detail in the next sections. Here
it is sucient to note that while forming I^ using only object
space computation (see section 4) is computationally expensive, we can compute D at the same time for little additional
computation. Also the derivative matrix is sparse with the
number of nonzero entries a few times the number of model
parameters. This makes the process described above a practical one.
The log-posterior is potentially multi-modal, and so it is important to begin the optimization from a good initialization.
where I^f p (~z; ~ ) denotes the pixel intensities in the image f
synthesized from the model, e2 is the noise variance and the
summation is over the pixels (p) and over all images (f ) used
for the inference.
Consider the negative log-posterior.
P
where D is the matrix of derivatives evaluated at z0 ; 0 . Then
the minimization of L(~z; ~) is replaced by minimization of the
quadratic form:
(6)
where the summation is over the individual measurement
points Ll , and L^ l (~z; ~) denotes the laser altimeter measurements synthesized from the model. The parameter l2 is the
variance of the laser altimeter measurement system.
We also assume that the images If comprising the data are
conditionally independent, giving
P
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>
use of image-space algorithms, with the fundamental assumption that each triangle making up the surface, when projected
onto the image plane, is much larger than a pixel. This makes
reasonable the assumption that any given pixel receives light
from only one triangle, but does produce images with artifacts
at the triangle edges. Standard rendering also produces inaccurate images if the triangles project into areas much smaller
than a pixel on the image plane, as the pixel will then be
colored with a value coming from just one of the triangles.
Clearly this approach is not suitable for high-resolution 3D
surface reconstruction from multiple images. The triangles
in a high-resolution surface may project onto an area much
smaller than a single pixel in the image plane (sub-pixel resolution). Therefore, as discussed in the introduction, for
our system we implemented a renderer for triangular meshes
which performs all computation in object space. At present
we neglect the blurring e ect due to di raction and due to
the role of pixel boundaries in the CCD array. Then the light
from a triangle as it is projected into a pixel contributes to
the brightness of the pixel with a weight factor proportional
to the fraction of the area of the triangle which projects into
that pixel. This produces anti-aliased images and allows an
image of any resolution to be produced from a mesh of arbitrary density, as required when the system performing the
surface inference may have no control over the image data
gathering.
Our renderer computes brightness I^p of a pixel p in the image
as a sum of contributions from individual surface triangles
4 whose projections into the image plane overlap, at least
partially, with the pixel p.
>
e cα
nα
θ
>
θ
es
v
α
p
i2
s
α
φ
α
108
p
i1
p
i0
Figure 2: Geometry of the triangular facet, illumination direction and viewing direction. ^zs is the vector to the illumination
source; z^v is the viewing direction.
In order to do this, the high resolution surface estimation
proceeds as follows.
1. Use a spline interpolation of the laser altimeter measurements to produce an initial height eld estimate at
the desired high resolution.
Also produce the matrix ^ ;l 1 , with zeros everywhere
except those diagonal entries corresponding to the
points of the high resolution surface grid that are measured by the laser altimeter. These values are 1=e2 .
Initialism all the albedos to 0:5.
2. Render the surface generated in step 1 and compute
the derivative matrices D (one for each image). Set
to zero all the derivatives with respect to the surface
heights. This xes the heights at their current values
in the optimization step (below), so that only the surface albedo values are inferred.
Starting from the surface from step 1, and using the
derivative matrices calculated above, use the conjugate
gradient algorithm to minimize the linearization of the
log-posterior in equation 9. This produces a good initialization for the nal optimization.
3. Render the surface generated in step 2 and compute
the derivative matrices D.
Starting from the surface in step 2, use the conjugate
gradient algorithm to minimize the linearization of the
log-posterior.
4. Repeat step 3 until convergence.
The minimum found is the nal surface estimate, which
combines the information from the laser altimeter measurements and the visible images.
I^p =
X
4
f4p 4 :
(10)
Here 4 is a radiation ux re ected from the triangular facet
4 and received by the camera, and f4p is the fraction of the
ux that falls onto a given pixel p in the image plane. In the
case of Lambertian surfaces and a single spectral band 4 is
given by the expression
4 = E ( s ) cos v cos ;
E ( s ) = A (I s cos s + I a ) :
= S=d2 :
(11)
Here is an average albedo of the triangular facet. Orientation angles s and v are de ned in gure 2. E ( s ) is the
total radiation ux incident on the triangular facet with area
A. This ux is modeled as a sum of two terms. Thes rst
term corresponds to direct radiation with intensity I from
the light source at in nity (commonly the sun). The second
term corresponds to ambient light with intensity I a . The parameter in equation. (11) is the angle between the camera
axis and the viewing direction (the vector from the surface
to the camera); is the lens fallo factor. in (11) is the
spatial angle subtended by the camera which is determined
by the area of the lens S and the distance d from the centroid
of the triangular facet to the camera.
We identify the triangular facet 4 by the set of 3 indices
(i0 ; i1 ; i2 ) from the vector of heights (1) that determines the
vertices of the triangle in a counterclockwise direction (see
gure 2). In the r.h.s of equation (11) we have omitted
for brevity those indices from all the quantities associated
with individual triangles. The average value of albedo for the
4 FORMATION OF THE IMAGE AND THE
DERIVATIVE MATRIX.
The task of forming an image, I^, given a surface description, ~z; ~, and camera and illumination parameters is the
area of computer graphics known as rendering (Foley 1990).
Most current rendering technology is focused on producing
images which are visually appealing, and producing them very
quickly. As discussed in the introduction, this results in the
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109
p
i1
L’
M
L
K’
pixel
K
δp
J’
p
i0
J
N
I’
I
p
i2
Figure 3: The intersection of the projection of a triangular
surface element (i0 ; i1 ; i2 ) onto the pixel plane with the pixel
boundaries. Bold lines corresponds to the edges of the polygon resulting from the intersection. Dashed lines correspond
to the new positions of the triangle edges when point Pi0 is
displaced by P
Figure 4: The initial synthetic surface model (Duckwater,
NV).
triangle in (11) is computed based on the components of the
albedo vector corresponding to the triangle indices
4.1 Computation of the derivative matrix.
4 i0 ;i1 ;i2 = 31 (i0 + i1 + i2 ):
Here A4 is the area of the projected triangle on the image
plane and Apolygon is the area of the polygon resulting from
the intersection of the projected triangle and boundary of the
pixel p (see gure 3).
The inference of the surface model parameters depends on
the ability to compute the derivatives of the modeled observations I^ with respect to the model parameters. According
to equation (10), the intensity I^p of a pixel p depends on the
subset of the surface parameters, (heights and albedos), that
are associated with the triangles whose projections overlap
the pixel area.
The derivatives I^p with respect to logarithmically transformed
albedo values are easily derived from equations (5), (10) and
(11).
In our object-space renderer, which is based on pixel-triangle
geometrical intersection in the image plane, the pixel intensity derivatives with respect to the surface heights have two
distinct contributions
(12)
We note that using average albedo (12) in the expression for
4 is an approximation which is justi ed when the albedo
values vary smoothly between the neighboring vertices of a
grid.
The area A of the triangle and the orientation angles in (11)
can be calculated in terms of the vertices of the triangle Pi
(see gure 2) as follows:
n^ ^zs = cos s ; n^ z^v = cos v ;
(13)
n^ = vi0 ;i1 2Avi1 ;i2 ; vi;j = Pj ; Pi
Here n^ is a unit normal to the triangular facet and vectors of
the edges of the triangle vi;j are shown in gure 2.
We use a standard pinhole camera model with no distortion in
which coordinates of a 3D world point P = (x; y; z) are rst
^ and then translated by the
rotated with the rotation matrix R
vector T into camera coordinates, yielding Pc = (xc ; yc ; zc)
Pc = R^ P + T
(14)
^
(R and T are expressed in terms of the camera registration
parameters (Hartley 2000). We do not give them explicitly
here). After the 3D transformation given in (14), point Pc
in the camera coordinate system is transformed using a perspective projection into the 2D image point P = (x; y) using
a focal length f and aspect ratio a.
x = ; f
y
zc
a xc :
yc
@ I^p = X f p @ 4 + @f4p
(17)
4 @z
@zi 4 4 @zi
i
Variation of the surface height zi gives rise to variations in
the normals of the triangles associated with this height (in a
general triangular mesh, on average 6 triangles are associated
with each height) and this produces the derivatives of the total radiation ux 4 to the camera from those triangles. This
is the rst term in equation (17). Also, height variation gives
rise to the displacement of the corresponding point which is
the projection of this vertex on the image plane. This results
in changes to the areas of the triangles and polygons with
edges containing this point (see gure 3). This produces the
derivatives of the fractions f4p , the second term in equation
17. Details of these derivatives can be found in (Smelyanskiy
2000, Morris 2001, Smelyanskiy 2001).
(15)
We use 2D image projections of the triangular vertices Pi to
compute the area fraction factors f4p for the surface triangles
(cf. Eq. (10))
A
(16)
f4p = polygon
:
5 RESULTS
Figure 4 shows the synthetic surface that we will use to
demonstrate our methodology. The topography is taken from
the USGS DEM of Duckwater, Nevada. A LANDSAT-TM
A4
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camera
image 1 look at
view up
camera
image 2 look at
view up
(75; 150; 2000)
(150; 150; 0)
(0; 1; 0)
(225; 150; 2000)
(150; 150; 0)
(0; 1; 0)
110
8
7.5
7
6.5
6
5.5
5
4.5
4
10
Table 1: Camera parameters used to generate the images in
gure 5
8
6
4
2
0
1
2
3
4
5
6
7
8
9
Figure 6: Grid from the 9 9 simulated laser altimeter measurements.
Figure 5: Images of the synthetic surface
image was co-registered with the DEM, and the values of
one band were used in place of the true albedos. This results
in the surface shown. One unit is approximately 180 meters.
Figure 5 shows two images rendered from the surface, and
table 1 gives the positions and orientations of the (synthetic)
cameras. The cameras were positioned to approximate satellite observations. The two images look very realistic. Note
that the image appear very similar due to the proximity of the
two camera positions. There is limited geometric information
available from the images alone.
Figure 6 shows a surface from a grid of 9 9 points extracted
from the surface in gure 4. The major terrain features have
all been sampled, but clearly it is a very poor representation of
the surface. This is taken as the laser altimeter observations
of the surface.
Using the images and the 9 9 grid, we will now go through
the surface estimation procedure that was detailed above.
Figure 7 shows the result of using the standard spline interpolation to expand the 9 9 grid to the full resolution of the
surface. The result is a smooth surface showing the major
features, but note that it contains no more information than
the coarse surface.
Keeping the heights xed at this surface, we then use the images to infer initial values for the albedos. The result is shown
in gure 8. Note that this is not simply the back-projection
of the of the images onto the surface, The information in
both images has been optimally combined to give the albedo
estimates. This surface is now a passable approximation to
the original surface, as it has high resolution albedo information providing rich visual detail, but clearly it contains no
topographic detail.
Figure 9 shows the nal inferred surface. This is much improved over the surface in gure 8. It shows that much of
the detail of the topography has been extracted from the data
and incorporated into the model. The error surfaces show in
gures 10 and 11 show clearly the improvement in the surface
estimate. These error surfaces show both the height error and
the albedo error as a shaded surface { the topography shows
the height error, and the colour of the surface shows the
Figure 7: Spline interpolation of the synthetic laser altimeter
measurements.
albedo error. The rms errors for the interpolated surface are
0:64 (per vertex) for the heights and 2:6 10;5 and for the
nal infered surface are 0:01 for the heights and 1:3 10;5
for the albedos. Note that the albedo values from the initialization are already quite good (as can be seen on gure 8,
however the inference process produces a topography which
is very signi cantly more accurate.
6 CONCLUSIONS AND FUTURE EXTENSIONS
We have presented the theory and practice of using Bayesian
methodology to combine the information in laser altimeter
measurements and visible images into a single, high resolution
surface model. We have shown on synthetic data that the two
data sets can be combined into a single high resolution model
that is more detailed than could be provided by either data
stream alone.
Current work is proceeding towards applying the demonstration system to real data, including NASA mission data. Work
in this area is devoted to sensor modeling (producing the synthetic images and derivative matrices for the actual imaging
sensor, rather than an idealization of it), estimation of the
camera positions to sub-pixel accuracy, better control of the
smoothness prior on the surface, and better initialization of
the optimization procedure.
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111
Figure 11: Error surface for the inferred surface
Figure 8: Interpolated surface with inferred albedos.
REFERENCES
Bernardo, J. and Smith, A., 1994. Bayesian Theory. Wiley,
Chichester, New York.
Foley, J., van Dam, A., Finer, S. and Hughes, J., 1990. Computer Graphics, Principles and Practice. Addison-Wesley, 2nd
edition.
Hartley, R. and Zisserman, A., 2000. Multiple View Geometry
in Computer Vision. Cambridge University Press.
Morris, R.D., Smelyanskiy, V.N., Cheeseman, P., 2001.
Matching Images to Models { Camera Calibration for 3-D
Surface Reconstruction. Proceedings of EMMCVPR 2001,
Springer LNCS volume 2134.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery,
B.P., 1992. Numerical Recipes in C. Cambridge University
Press, 2nd edition.
Rees, W., 1990. Physical Principles of Remote Sensing.
Cambridge University Press.
Smelyanskiy, V.N., Cheeseman, P., Maluf, D.A., Morris,
R.D., Bayesian Super-Resolved Surface Reconstruction from
Images. Proceedings of Computer Vision and Pattern Recognition 2000.
Smelyanskiy, V.N., Morris, R.D., Maluf, D.A., Cheeseman,
P., 2001. (Almost) Featureless Stereo { Calibration and
Dense 3D Reconstruction Using Whole Image Operations.
Technical Report, RIACS, NASA Ames Research Center.
Figure 9: Final inferred surface using the Bayesian approach
to combining the laser altimeter and visible images
Figure 10: Error surface for the interpolated surface
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